Abstract
This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.
Highlights
Lyapunov’s method, which makes an essential use of auxiliary functions, is an important approach to study the stability of differential systems including ordinary differential equations (ODEs) and stochastic differential equations (SDEs)
Lyapunov’s method is modified to the study of stability of Markovian processes [4], stochastic differential systems based on Brownian motions [5], semimartingales [6], or Levy processes [7] and is developed with the form of exponential stability [8] or LaSalle theorem [9], and so forth
We extend the stability criteria of two measures to the mean stability situations for the stochastic systems with uncertainty
Summary
Lyapunov’s method, which makes an essential use of auxiliary functions ( called Lyapunov functions), is an important approach to study the stability of differential systems including ordinary differential equations (ODEs) and stochastic differential equations (SDEs). This method started in Lyapunov’s original work in 1892 [1] for demonstrating stability of ODEs. In the 1960s, Movchan [2] studied the stability with two measures; such works were developed and can be seen in [3]. We extend Lyapunov’s methods used by [3] for ODEs to the stochastic cases and study the mean stability criteria in terms of two measures for system (1).
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